Tectonic activity through time transforms an initially uniform stratified terrain composed of a continuous stack of substantially level surfaces into an uneven terrain that may be eroded, affected by shifts in sedimentary deposition patterns, folded, and/or fractured by faults forming discontinuities across the originally continuous horizons. To model the original past time of deposition, referred to as “geological time”, from data collected from the current present-day subsurface structures (e.g., to “reverse time”), a depositional model (e.g. referred to as a “GeoChron” model) may simulate a reversal of such erosion and tectonic activity.
“Geological horizons” are identical or approximate to level-sets of the geological-time. As a consequence, modeling or refining the geological-time may be equivalent to modeling or refining the horizons. The actual geological-time “t” may equivalently be replaced by a given continuous strictly monotonic function F(t) (e.g. a function whose 1st derivative never reduces to zero, or a function that is either strictly decreasing or strictly increasing) of the actual geological-time. Such a transformation typically does not change the geometry of the level-sets (e.g. geological horizons). Thus, “geological time” may refer to any continuous strictly monotonic function of the actual or predicted geological time. In the following, as an example provided for the sake of clarity, the geological time may be assumed to be strictly monotonically increasing (e.g. more recent deposited or top layer subsurface particles being relatively younger, such as, deposited at a geological time of 4.5 billion years, than deeper subsurface particles, such as, deposited at a geological time of 4.2 billion years). From a physical perspective, a geological time function that is strictly monotonically increasing may be equivalent to time never stopping and/or never running backwards. Equivalently, the geological time function may be strictly monotonically decreasing. In such a case, all the inequalities referring directly or indirectly to geological time (e.g., equations 14 and 15) may be inverted.
Each particle of sediment observed today in the subsurface was originally deposited at a paleo-geographic location (u,v) and a geologic time (t). The set of particles of sediment sharing a common paleo-geographic location is called an “Iso Paleo Geographic” (IPG) line which consists of a curve approximately orthogonal to the geologic horizons. There are several techniques known in the art to build these IPG lines. According to embodiments of the invention, any point located in the subsurface may be intersected by one (and only one) unique IPG-line.
Generally speaking, depositional models may be generated by applying 3D interpolation techniques to the current time models to determine the geological time throughout the entire sampled volume. Current interpolation techniques for generating depositional models typically use extensive simplifications that often violate for example principles of superposition and minimal energy deformations, thereby rendering inaccurate data. Current interpolation techniques used thus far incorrectly assume that the gradient (e.g. multi-dimensional or directional vector, slope or derivative) of the 3D geological time function t is continuous everywhere within each stratigraphic sequence contained within each fault block, and in particular across some reference horizons.
FIG. 1 shows a vertical cross-section of a present-day geological model where the variations of the geological time function 190 are represented by a grayscale color map. In FIG. 1, the gradient of the geological time function is discontinuous across a horizon 150 (the white curve). The discontinuity of the gradient is particularly visible within regions encircled by ellipses 160. The geological time function 190 across the horizon is C0 (its 0th derivative, i.e., the function itself, is continuous), but not C1 (its first derivative is not continuous). Curve 140 is an IPG-line approximately orthogonal to the horizon (level-set) of the geological time function. Along the IPG-line 140, the spacing (gradient) of the level sets between points 110 and 120 is different from the spacing (gradient) between points 120 and 130. This shows that the geological time function 190 is not C1 at point 120.
FIG. 2 shows the variations of the geological time function 290 (e.g. 190 in FIG. 1), for example, as a 1D function of the curvilinear abscissa along a curve 240 (e.g. function of location along the 1D line 140 in FIG. 1). In practice, the geological time may be defined or measured at a plurality of sampling points, for example, scattered in the geological domain. These sampling points may be used to approximate (e.g. estimate) the geological time as a 3D function everywhere in the geological domain. When the geological domain is traversed by a 1D line 240, the 3D function representing the geological time may be represented by a 1D function of the curvilinear abscissa along this 1D line. For example, in FIG. 2, the vertical axis 290 represents the continuously (e.g. without gaps) interpolated geological time and the horizontal axis 240 represents the curvilinear abscissa along the 1D line 140. The black curve 280 corresponds to a classical C1 interpolation (e.g. where the 1st order derivative is continuous) of the geological time function (e.g. an interpolation that results in a geological time) between the points 210 (110 in FIG. 1) and 230 (130 in FIG. 1). The gray curve 270 corresponds to a C0 piecewise linear interpolation (e.g. composed of adjacent straight-line segments) of the geological time function between points 210-220 (110-120 in FIG. 1) and 220-230 (120-130 in FIG. 1). In the neighborhood of point 220 (120 in FIG. 1), the classical C1 interpolation generates a geological time that is not strictly monotonic and oscillates (e.g. increasing and decreasing) along the path of the 1D line 240 (140 in FIG. 1). These oscillations, referred to as the “Gibbs effect,” cause a zero gradient (e.g. slope of the curve 280) at the peaks (maxima) 281 and troughs (minima) 282 where the geological-time function is non-monotonic. This non-monotonic behavior of the geological-time function is physically unlikely or impossible because the higher a particle of sediment is located in the stratigraphic column (e.g. along path 240 in FIG. 2 or 140 in FIG. 1), the later the geological time when it was deposited in the Earth. A non-monotonic geological-time function may have level-set surfaces that are closed surfaces, which appear as “bubbles” in the model, and which generally correspond to a physically unacceptable geometry for a geologic horizon. In order to avoid generating such bubbles, a common practice of classical C1 interpolators is to strongly smooth the variations of the geological time function, as illustrated by curve 250. Unfortunately such severe smoothing causes the observed sampling points to be incorrectly fitted. As a consequence, there is a need inherent in the art for “refining” an initial strictly monotonic 3D function approximating the geological time function in order to accurately model geological horizon surfaces, particularly in areas in which the gradient of the geological-time function is discontinuous.